1 DEFINITION OF A MEASURE 101 a linear form on tf (X). It is a measure, for if K is any compact subset of X and if /e Jf (X; K), then \gf\ £ \\f\\ • sup \g(x)\9 and consequently I ^ bK II/H, where fcK = % sup |#(*)|. This measure is denoted by g • xeK and it is called the measure with density g relative to ju (cf. (13.13)). (13.1.6) Let TC : X-»X' be a homeomorphism of X onto a locally compact space X'. For each function /e «?f(X'), the function /° n belongs to ^f(X), and we have Supp(/o n) = n"1(Suppf). It follows immediately that, if \L is any measure on X, then/W/i(/o 71) is a measure on X', called the zma#e of /i under TC and denoted by 7t(/j). (1 3,1 .7) Let Y be a closed subset of X (and therefore a locally compact sub- space of X (3.18.4)) and v a measure on Y. For each / e Jf (X; K), the restriction /| Y belongs to Jf(Y; K n Y), and hence there exists a constant CK such that yeY for all/e ^(X; K). The mapping /V-»v(/|Y) is therefore a measure on X, called the *>wa#£ of v under the canonical injection Y -> X, or the canonical extension ofv to X. (1 3.1 .8) Let U be an open subset of X (and therefore again a locally compact subspace (3.18.4)). For each compact subset K of U, it is clear that the mapping/W/| U is an isometry of Jf (X; K) onto tf (U ; K). The image under the inverse isometry of a function ge Jf(U; K) is the function #u which agrees with g on U and is zero on X - U. (By abuse of notation, we shall often write /in place of/| U when Supp(/) c U, and g in place of #u). The mapping g^g" of Jfc(U) into ^fc(X) is therefore injective. If /* is any measure on X, the mapping g h-» X0U) is a measure on U, said to be induced by ju OT U, or the restriction of p. to U, and denoted by ^ or ju | U. It should be noted that a measure v on U is not necessarily the restriction of a measure on X (Section (13.4), Problem 1), and that an "extension" of v to X, if it exists, is not in general unique. However, there is the following result: (13.1.9) Let (Ua)a6l be an open covering ofX. For each a el let ua be a measure on Ua such that, for each pair of indices a, j8, the restrictions of i*a and Up to Ua n U^ (13.1 .8) are equal. Then there exists a unique measure \i on X whose restriction to Ua is na,for each a 6 1. We shall first show that each function /e tf C(X) can be written in the form/= £/, where, for each index i, there exists a( e I such that/f e Jfc(X)ntable set of points of discontinuity of / is dense in [a, b[ (Section 3.15,